In chaos theory, the problem anti-synchronization of chaotic systems deals with a pair of chaotic systems called drive and response systems. In this problem, the design goal is to drive the sum of their respective states to zero asymptotically. This problem gets even more complicated and requires special attention when the parameters of the drive and response systems are unknown. This paper uses adaptive control theory and Lyapunov stability theory to derive new results for the anti-synchronization of hyperchaotic Wang system (2008) and hyperchaotic Li system (2005) with uncertain parameters. Hyperchaotic systems are nonlinear dynamical systems exhibiting chaotic behaviour with two or more positive Lyapunov exponents. The hyperchaotic systems have applications in areas like oscillators, lasers, neural networks, encryption, secure transmission and secure communication. The main results derived in this paper are validated and demonstrated with MATLAB simulations.

1. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013
DOI : 10.5121/ijcseit.2013.3203 31
ANTI-SYNCHRONIZATION OF HYPERCHAOTIC
WANG AND HYPERCHAOTIC LI SYSTEMS WITH
UNKNOWN PARAMETERS VIA ADAPTIVE CONTROL
Sundarapandian Vaidyanathan1
1
Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University
Avadi, Chennai-600 062, Tamil Nadu, INDIA
sundarvtu@gmail.com
ABSTRACT
In chaos theory, the problem anti-synchronization of chaotic systems deals with a pair of chaotic systems
called drive and response systems. In this problem, the design goal is to drive the sum of their respective
states to zero asymptotically. This problem gets even more complicated and requires special attention when
the parameters of the drive and response systems are unknown. This paper uses adaptive control theory
and Lyapunov stability theory to derive new results for the anti-synchronization of hyperchaotic Wang
system (2008) and hyperchaotic Li system (2005) with uncertain parameters. Hyperchaotic systems are
nonlinear dynamical systems exhibiting chaotic behaviour with two or more positive Lyapunov exponents.
The hyperchaotic systems have applications in areas like oscillators, lasers, neural networks, encryption,
secure transmission and secure communication. The main results derived in this paper are validated and
demonstrated with MATLAB simulations.
KEYWORDS
Hyperchaos, Hyperchaotic Systems, Adaptive Control, Anti-Synchronization.
1. INTRODUCTION
Hyperchaotic systems are typically defined as nonlinear chaotic systems having two or more
positive Lyapunov exponents. They are applicable in several areas like lasers [1], chemical
reactions [2], neural networks [3], oscillators [4], data encryption [5], secure communication [6-
8], etc.
In chaos theory, the anti-synchronization problem deals with a pair of chaotic systems called the
drive and response systems, where the design goal is to render the respective states to be same in
magnitude, but opposite in sign, or in other words, to drive the sum of the respective states to zero
asymptotically [9].
There are several methods available in the literature to tackle the problem of synchronization and
anti-synchronization of chaotic systems like active control method [10-12], adaptive control
method [13-15], backstepping method [16-19], sliding control method [20-22] etc.
This paper derives new results for the adaptive controller design for the anti-synchronization of
hyperchaotic Wang systems ([23], 2008) and hyperchaotic Li systems ([24], 2005) with unknown
parameters. Lyapunov stability theory [25] has been applied to prove the main results of this
paper. Numerical simulations have been shown using MATLAB to illustrate the results.

2. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013
32
2. THE PROBLEM OF ANTI-SYNCHRONIZATION OF CHAOTIC SYSTEMS
In chaos synchronization problem, the drive system is described by the chaotic dynamics
( )x Ax f x= + (1)
where A is the n n× matrix of the system parameters and : n n
f →R R is the nonlinear part.
Also, the response system is described by the chaotic dynamics
( )y By g y u= + + (2)
where B is the n n× matrix of the system parameters, : n n
g →R R is the nonlinear part and
n
u ∈R is the active controller to be designed.
For the pair of chaotic systems (1) and (2), the design goal of the anti-synchronization problem is
to construct a feedback controller ,u which anti-synchronizes their states for all (0), (0) .n
x y ∈R
The anti-synchronization error is defined as
,e y x= + (3)
The error dynamics is obtained as
( ) ( )e By Ax g y f x u= + + + + (4)
The design goal is to find a feedback controller uso that
lim ( ) 0
t
e t
→∞
= for all (0)e ∈Rn
(5)
Using the matrix method, we consider a candidate Lyapunov function
( ) ,T
V e e Pe= (6)
where P is a positive definite matrix.
It is noted that : n
V →R R is a positive definite function.
If we find a feedback controller u so that
( ) ,T
V e e Qe= − (7)
where Q is a positive definite matrix, then : n
V → R R is a negative definite function.
Thus, by Lyapunov stability theory [26], the error dynamics (4) is globally exponentially stable.
When the system parameters in (1) and (2) are unknown, we apply adaptive control theory to
construct a parameter update law for determining the estimates of the unknown parameters.

3. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013
33
3. HYPERCHAOTIC WANG AND HYPERCHAOTIC LI SYSTEMS
The hyperchaotic Wang system ([23], 2008) is given by
1 2 1 2 3
2 1 1 3 2 4
3 3 1 2
4 4 1 3
( )
0.5
0.5
x a x x x x
x cx x x x x
x dx x x
x bx x x
= − +
= − − −
= − +
= +




(8)
where , , ,a b c d are constant, positive parameters of the system.
The Wang system (8) depicts a hyperchaotic attractor for the parametric values
40, 1.7, 88, 3a b c d= = = = (9)
The Lyapunov exponents of the system (8) are determined as
1 2 3 43.2553, 1.4252, 0, 46.9794   = = = = − (10)
Since there are two positive Lyapunov exponents in (10), the Wang system (8) is hyperchaotic for
the parametric values (9).
Figure 1 shows the phase portrait of the hyperchaotic Wang system.
The hyperchaotic Li system ([24], 2005) is given by
1 2 1 4
2 1 1 3 2
3 3 1 2
4 2 3 4
( )x x x x
x x x x x
x x x x
x x x rx

 

= − +
= − +
= − +
= +




(11)
where , , , ,r    are constant, positive parameters of the system.
The Li system (11) depicts a hyperchaotic attractor for the parametric values
35, 3, 12, 7, 0.58r   = = = = = (12)
The Lyapunov exponents of the system (11) for the parametric values in (12) are
1 2 3 40.5011, 0.1858, 0, 26.1010   = = = = − (13)
Since there are two positive Lyapunov exponents in (13), the Li system (11) is hyperchaotic for
the parametric values (12). Figure 2 shows the phase portrait of the hyperchaotic Li system.

4. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013
34
Figure 1. Hyperchaotic Attractor of the Hyperchaotic Wang System
Figure 2. Hyperchaotic Attractor of the Hyperchaotic Li System
4. ANTI-SYNCHRONIZATION OF HYPERCHAOTIC WANG SYSTEMS VIA
ADAPTIVE CONTROL
In this section, we derive new results for designing a controller for the anti-synchronization of
identical hyperchaotic Wang systems (2008) with unknown parameters via adaptive control.
The drive system is the hyperchaotic Wang dynamics given by

5. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013
35
1 2 1 2 3
2 1 1 3 2 4
3 3 1 2
4 4 1 3
( )
0.5
0.5
x a x x x x
x cx x x x x
x dx x x
x bx x x
= − +
= − − −
= − +
= +




(14)
where , , ,a b c d are unknown parameters of the system and 4
x∈ R is the state.
The response system is the controlled hyperchaotic Wang dynamics given by
1 2 1 2 3 1
2 1 1 3 2 4 2
3 3 1 2 3
4 4 1 3 4
( )
0.5
0.5
y a y y y y u
y cy y y y y u
y dy y y u
y by y y u
= − + +
= − − − +
= − + +
= + +




(15)
where 4
y ∈ R is the state and 1 2 3 4, , ,u u u u are the adaptive controllers to be designed.
For the anti-synchronization, the error e is defined as
1 1 1 2 2 2 3 3 3 4 4 4, , ,e e e ey x y x y x y x= + = + = + = + (16)
Then we derive the error dynamics as
1 2 1 2 3 2 3 1
2 1 2 4 1 3 1 3 2
3 3 1 2 1 2 3
4 4 1 3 1 3 4
( )
0.5
0.5( )
e a e e y y x x u
e ce e e y y x x u
e de y y x x u
e be y y x x u
= − + + +
= − − − − +
= − + + +
= + + +




(17)
The adaptive controller to solve the anti-synchronization problem is taken as
1 2 1 2 3 2 3 1 1
2 1 2 4 1 3 1 3 2 2
3 3 1 2 1 2 3 3
4 4 1 3 1 3 4 4
ˆ( )( )
ˆ( ) 0.5
ˆ( )
ˆ( ) 0.5( )
u a t e e y y x x k e
u c t e e e y y x x k e
u d t e y y x x k e
u b t e y y x x k e
= − − − − −
= − + + + + −
= − − −
= − − + −
(18)
In Eq. (18), , ( 1,2,3,4)ik i = are positive gains and ˆ ˆˆ ˆ( ), ( ), ( ), ( )a t b t c t d t are estimates for the
unknown parameters , , ,a b c d respectively.
By the substitution of (18) into (17), the error dynamics is obtained as
1 2 1 1 1
2 1 2 2
3 3 3 3
4 4 4 4
ˆ( ( ))( )
ˆ( ( ))
ˆ( ( ))
ˆ( ( ))
e a a t e e k e
e c c t e k e
e d d t e k e
e b b t e k e
= − − −
= − −
= − − −
= − −




(19)

6. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013
36
Next, we define the parameter estimation errors as
ˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )a b c de t a a t e t b b t e t c c t e t d d t= − = − = − = − (20)
Upon differentiation, we get
ˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )a b c de t a t e t b t e t c t e t d t= − = − = − = −
      (21)
Substituting (20) into the error dynamics (19), we obtain
1 2 1 1 1
2 1 2 2
3 3 3 3
4 4 4 4
( )a
c
d
b
e e e e k e
e e e k e
e e e k e
e e e k e
= − −
= −
= − −
= −




(22)
We consider the candidate Lyapunov function
( )2 2 2 2 2 2 2 2
1 2 3 4
1
2
a b c dV e e e e e e e e= + + + + + + + (23)
Differentiating (23) along the dynamics (21) and (22), we obtain
( )
( ) ( )
2 2 2 2 2
1 1 2 2 3 3 4 4 1 2 1 4
2
1 2 3
ˆˆ( )
ˆˆ
a b
c d
V k e k e k e k e e e e e a e e b
e e e c e e d
 = − − − − + − − + − 
+ − + − −


(24)
In view of (24), we choose the following parameter update law:
1 2 1 5
2
4 6
1 2 7
2
3 8
ˆ ( )
ˆ
ˆ
ˆ
a
b
c
d
a e e e k e
b e k e
c e e k e
d e k e
= − +
= +
= +
= − +




(25)
Next, we prove the following main result of this section.
Theorem 4.1 The adaptive control law defined by Eq. (18) along with the parameter update law
defined by Eq. (25), where ,( 1,2, ,8)ik i =  are positive constants, render global and exponential
anti-synchronization of the identical hyperchaotic Wang systems (14) and (15) with unknown
parameters for all initial conditions 4
(0), (0) .x y ∈ R In addition, the parameter estimation errors
( ), ( ), ( ), ( )a b c de t e t e t e t globally and exponentially converge to zero for all initial conditions.

7. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013
37
Proof. The proof is via Lyapunov stability theory [25] by taking V defined by Eq. (23) as the
candidate Lyapunov function. Substituting the parameter update law (25) into (24), we get
2 2 2 2 2 2 2 2
1 1 2 2 3 3 4 4 5 6 7 8( ) a b c dV e k e k ek e k e k e k e k e k e= − −− − − − − − (26)
which is a negative definite function on
8
.R This completes the proof. 
Next, we demonstrate our adaptive anti-synchronization results with MATLAB simulations. The
classical fourth order R-K method with time-step 8
10h −
= has been used to solve the hyperchaotic
Wang systems (14) and (15) with the adaptive controller defined by (18) and parameter update
law defined by (25).
The feedback gains in the adaptive controller (18) are taken as 5, ( 1, ,8).ik i= = 
The parameters of the hyperchaotic Wang systems are taken as in the hyperchaotic case, i.e.
40, 1.7, 88, 3a b c d= = = =
For simulations, the initial conditions of the drive system (14) are taken as
1 2 3 4(0) 37, (0) 16, (0) 14, (0) 11x x x x= = − = =
Also, the initial conditions of the response system (15) are taken as
1 2 3 4(0) 21, (0) 32, (0) 28, (0) 8y y y y= = = − = −
Also, the initial conditions of the parameter estimates are taken as
ˆ ˆˆ ˆ(0) 12, (0) 4, (0) 6, (0) 5a b c d= = = − =
Figure 3 depicts the anti-synchronization of the identical hyperchaotic Wang systems.
Figure 4 depicts the time-history of the anti-synchronization errors 1 2 3 4, , , .e e e e
Figure 5 depicts the time-history of the parameter estimation errors , , , .a b c de e e e

8. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013
38
Figure 3. Anti-Synchronization of Identical Hyperchaotic Wang Systems
Figure 4. Time-History of the Anti-Synchronization Errors 1 2 3 4, , ,e e e e

9. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013
39
Figure 5. Time-History of the Parameter Estimation Errors , , ,a b c de e e e
5. ANTI-SYNCHRONIZATION OF HYPERCHAOTIC LI SYSTEMS VIA ADAPTIVE
CONTROL
In this section, we derive new results for designing a controller for the anti-synchronization of
identical hyperchaotic Li systems (2005) with unknown parameters via adaptive control.
The drive system is the hyperchaotic Li dynamics given by
1 2 1 4
2 1 1 3 2
3 3 1 2
4 2 3 4
( )x x x x
x x x x x
x x x x
x x x rx

 

= − +
= − +
= − +
= +




(27)
where , , , , r    are unknown parameters of the system and 4
x∈ R is the state.
The response system is the controlled hyperchaotic Li dynamics given by
1 2 1 4 1
2 1 1 3 2 2
3 3 1 2 3
4 2 3 4 4
( )y y y y u
y y y y y u
y y y y u
y y y ry u

 

= − + +
= − + +
= − + +
= + +




(28)
where 4
y ∈ R is the state and 1 2 3 4, , ,u u u u are the adaptive controllers to be designed.
For the anti-synchronization, the error e is defined as
1 1 1 2 2 2 3 3 3 4 4 4, , ,e e e ey x y x y x y x= + = + = + = + (29)

10. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013
40
Then we derive the error dynamics as
1 2 1 4 1
2 1 2 1 3 1 3 2
3 3 1 2 1 2 3
4 4 2 3 2 3 4
( )e e e e u
e e e y y x x u
e e y y x x u
e re y y x x u

 

= − + +
= + − − +
= − + + +
= + + +




(30)
The adaptive controller to solve the anti-synchronization problem is taken as
1 2 1 4 1 1
2 1 2 1 3 1 3 2 2
3 3 1 2 1 2 3 3
4 4 2 3 2 3 4 4
ˆ( )( )
ˆ ˆ( ) ( )
ˆ( )
ˆ( )
u t e e e k e
u t e t e y y x x k e
u t e y y x x k e
u r t e y y x x k e

 

= − − − −
= − − + + −
= − − −
= − − − −
(31)
In Eq. (31), , ( 1,2,3,4)ik i = are positive gains and ˆ ˆˆ ˆ ˆ( ), ( ), ( ), ( ), ( )t t t t r t    are estimates for
the unknown parameters , , , , r    respectively.
By the substitution of (31) into (30), the error dynamics is obtained as
1 2 1 1 1
2 1 2 2 2
3 3 3 3
4 4 4 4
ˆ( ( ))( )
ˆ ˆ( ( )) ( ( ))
ˆ( ( ))
ˆ( ( ))
e t e e k e
e t e t e k e
e t e k e
e r r t e k e
 
   
 
= − − −
= − + − −
= − − −
= − −




(32)
Next, we define the parameter estimation errors as
ˆ ˆˆ ˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )re t t e t t e t t e t t e t r r t          = − = − = − = − = − (33)
Upon differentiation, we get
ˆ ˆˆ ˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )re t t e t t e t t e t t e t r t      = − = − = − = − = −
        (34)
Substituting (33) into the error dynamics (32), we obtain
1 2 1 1 1
2 1 2 2 2
3 3 3 3
4 4 4 4
( )
r
e e e e k e
e e e e e k e
e e e k e
e e e k e

 

= − −
= + −
= − −
= −




(35)
We consider the candidate Lyapunov function

11. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013
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( )2 2 2 2 2 2 2 2 2
1 2 3 4
1
2
rV e e e e e e e e e   = + + + + + + + + (36)
Differentiating (36) along the dynamics (34) and (35), we obtain
( )
( ) ( ) ( )
2 2 2 2 2
1 1 2 2 3 3 4 4 1 2 1 3
2 2
2 1 2 4
ˆˆ( )
ˆˆ ˆr
V k e k e k e k e e e e e e e
e e e e e e e r
 
 
 
 
 = − − − − + − − + − − 
+ − + − + −

 
(37)
In view of (37), we choose the following parameter update law:
1 2 1 5
2
3 6
2
2 7
1 2 8
2
4 9
ˆ ( )
ˆ
ˆ
ˆ
ˆ
a
r
e e e k e
e k e
e k e
e e k e
r e k e







= − +
= − +
= +
= +
= +





(38)
Next, we prove the following main result of this section.
Theorem 5.1 The adaptive control law defined by Eq. (31) along with the parameter update law
defined by Eq. (38), where ,( 1,2, ,9)ik i =  are positive constants, render global and exponential
anti-synchronization of the identical hyperchaotic Li systems (27) and (28) with unknown
parameters for all initial conditions 4
(0), (0) .x y ∈ R In addition, the parameter estimation errors
( ), ( ), ( ), ( ), ( )re t e t e t e t e t    globally and exponentially converge to zero for all initial
conditions.
Proof. The proof is via Lyapunov stability theory [25] by taking V defined by Eq. (36) as the
candidate Lyapunov function. Substituting the parameter update law (38) into (37), we get
2 2 2 2 2 2 2 2 2
1 1 2 2 3 3 4 4 5 6 7 8 9( ) rV e k e k ek e k e k e k e k e k e k e   = − −− − − − − − − (39)
which is a negative definite function on
9
.R This completes the proof. 
Next, we demonstrate our adaptive anti-synchronization results with MATLAB simulations. The
classical fourth order R-K method with time-step 8
10h −
= has been used to solve the hyperchaotic
Li systems (27) and (28) with the adaptive controller defined by (31) and parameter update law
defined by (38). The feedback gains in the adaptive controller (31) are taken as
5, ( 1, ,9).ik i= = 
The parameters of the hyperchaotic Li systems are taken as in the hyperchaotic case, i.e.
35, 3, 12, 7, 0.58r   = = = = =
For simulations, the initial conditions of the drive system (27) are taken as

12. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013
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1 2 3 4(0) 7, (0) 26, (0) 12, (0) 14x x x x= = − = =
Also, the initial conditions of the response system (28) are taken as
1 2 3 4(0) 21, (0) 28, (0) 18, (0) 29y y y y= − = = − =
Also, the initial conditions of the parameter estimates are taken as
ˆ ˆˆ ˆ ˆ(0) 7, (0) 15, (0) 5, (0) 4, (0) 3r   = = = = = −
Figure 6 depicts the anti-synchronization of the identical hyperchaotic Li systems.
Figure 7 depicts the time-history of the anti-synchronization errors 1 2 3 4, , , .e e e e
Figure 8 depicts the time-history of the parameter estimation errors , , , , .re e e e e   
Figure 6. Anti-Synchronization of Identical Hyperchaotic Li Systems

13. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013
43
Figure 7. Time-History of the Anti-Synchronization Errors 1 2 3 4, , ,e e e e
Figure 8. Time-History of the Parameter Estimation Errors , , , , re e e e e   
6.ANTI-SYNCHRONIZATION OF HYPERCHAOTIC WANG AND
HYPERCHAOTIC LI SYSTEMS VIA ADAPTIVE CONTROL
In this section, we derive new results for designing a controller for the anti-synchronization of
non-identical hyperchaotic Wang system (2009) and hyperchaotic Li system (2005) with
unknown parameters via adaptive control.
The drive system is the hyperchaotic Wang dynamics given by

14. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013
44
1 2 1 2 3
2 1 1 3 2 4
3 3 1 2
4 4 1 3
( )
0.5
0.5
x a x x x x
x cx x x x x
x dx x x
x bx x x
= − +
= − − −
= − +
= +




(40)
where , , ,a b c d are unknown parameters of the system and 4
x ∈ R is the state.
The response system is the controlled hyperchaotic Li dynamics given by
1 2 1 4 1
2 1 1 3 2 2
3 3 1 2 3
4 2 3 4 4
( )y y y y u
y y y y y u
y y y y u
y y y ry u

 

= − + +
= − + +
= − + +
= + +




(41)
where , , , , r    are unknown parameters, 4
y ∈ R is the state and 1 2 3 4, , ,u u u u are the adaptive
controllers to be designed.
For the anti-synchronization, the error e is defined as
1 1 1 2 2 2 3 3 3 4 4 4, , ,e e e ey x y x y x y x= + = + = + = + (42)
Then we derive the error dynamics as
1 2 1 2 1 4 2 3 1
2 1 1 2 2 4 1 3 1 3 2
3 3 3 1 2 1 2 3
4 4 4 1 3 2 3 4
( ) ( )
0.5
0.5
e a x x y y y x x u
e cx y x y x x x y y u
e dx y y y x x u
e bx ry x x y y u

 

= − + − + + +
= + − + − − − +
= − − + + +
= + + + +




(43)
The adaptive controller to solve the anti-synchronization problem is taken as
1 2 1 2 1 4 2 3 1 1
2 1 1 2 2 4 1 3 1 3 2 2
3 3 3 1 2 1 2 3 3
4 4 4 1 3 2 3 4 4
ˆˆ( )( ) ( )( )
ˆ ˆˆ( ) ( ) ( ) 0.5
ˆ ˆ( ) ( )
ˆ ˆ( ) ( ) 0.5
u a t x x t y y y x x k e
u c t x t y x t y x x x y y k e
u d t x t y y y x x k e
u b t x r t y x x y y k e

 

= − − − − − − −
= − − + − + + + −
= + − − −
= − − − − −
(44)
In Eq. (44), , ( 1,2,3,4)ik i = are positive gains and ˆ( ),a t ˆ( ),b t ˆ( ),c t ˆ( ),d t ˆ( ),t ˆ( ),t ˆ( ),t
ˆ( ),t ˆ( )r t are estimates for the unknown parameters , , , , , , , ,a b c d r    respectively.
By the substitution of (44) into (43), the error dynamics is obtained as
1 2 1 2 1 1 1
2 1 1 2 2 2
3 3 3 3 3
4 4 4 4 4
ˆˆ( ( ))( ) ( ( ))( )
ˆ ˆˆ( ( )) ( ( )) ( ( ))
ˆ ˆ( ( )) ( ( ))
ˆ ˆ( ( )) ( ( ))
e a a t x x t y y k e
e c c t x t y t y k e
e d d t x t y k e
e b b t x r r t y k e
 
   
 
= − − + − − −
= − + − + − −
= − − − − −
= − + − −




(45)

15. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013
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Next, we define the parameter estimation errors as
ˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )
ˆ ˆˆ ˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )
a b c d
r
e t a a t e t b b t e t c c t e t d d t
e t t e t t e t t e t t e t r r t          
= − = − = − = −
= − = − = − = − = −
(46)
Upon differentiation, we get
ˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )
ˆ ˆˆ ˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )
a b c d
r
e t a t e t b t e t c t e t d t
e t t e t t e t t e t t e t r t      
= − = − = − = −
= − = − = − = − = −
     
       
(47)
Substituting (46) into the error dynamics (45), we obtain
1 2 1 2 1 1 1
2 1 1 2 2 2
3 3 3 3 3
4 4 4 4 4
( ) ( )a
c
d
b r
e e x x e y y k e
e e x e y e y k e
e e x e y k e
e e x e y k e

 

= − + − −
= + + −
= − − −
= + −




(48)
We consider the candidate Lyapunov function
( )2 2 2 2 2 2 2 2 2 2 2 2 2
1 2 3 4
1
2
a b c d rV e e e e e e e e e e e e e   = + + + + + + + + + + + + (49)
Differentiating (49) along the dynamics (47) and (48), we obtain
( ) ( )
( ) ( )
( ) ( ) ( )
2 2 2 2
1 1 2 2 3 3 4 4 1 2 1 4 4 2 1
3 3 1 2 1 3 3
2 2 2 1 4 4
ˆˆ ˆ( )
ˆ ˆˆ+ ( )
ˆˆ ˆ
a b c
d
r
V k e k e k e k e e e x x a e e x b e e x c
e e x d e e y y e e y
e e y e e y e e y r
 
 
 
 
 = − − − − + − − + − + − 
 − − + − − + − − 
+ − + − + −
 
 
 
(50)
In view of (50), we choose the following parameter update law:
1 2 1 5 1 2 1 9
4 4 6 3 3 10
2 1 7 2 2 11
3 3 8 2 1 12
ˆˆ ( ) , ( )
ˆ ˆ,
ˆˆ ,
ˆ ˆ,
a a
b
c
d
a e x x k e e y y k e
b e x k e e y k e
c e x k e e y k e
d e x k e e y k






= − + = − +
= + = − +
= + = +
= − + = +

 

 
4 4 13
ˆ r
e
r e y k e

= +
(51)
Next, we prove the following main result of this section.

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Theorem 6.1 The adaptive control law defined by Eq. (44) along with the parameter update law
defined by Eq. (51), where ,( 1,2, ,13)ik i =  are positive constants, render global and
exponential anti-synchronization of the non-identical hyperchaotic Wang system (40) and
hyperchaotic Li system (41) with unknown parameters for all initial conditions 4
(0), (0) .x y ∈ R
In addition, all the parameter estimation errors globally and exponentially converge to zero for all
initial conditions.
Proof. The proof is via Lyapunov stability theory [25] by taking V defined by Eq. (49) as the
candidate Lyapunov function. Substituting the parameter update law (51) into (50), we get
2 2 2 2 2 2 2 2
1 1 2 2 3 3 4 4 5 6 7 8
2 2 2 2 2
9 10 11 12 13
( ) a b c d
r
V e k e k e
k e
k e k e k e k e k e k e
k e k e k e k e   
= − −
−
− − − − − −
− − − −

(52)
which is a negative definite function on
13
.R This completes the proof. 
Next, we demonstrate our adaptive anti-synchronization results with MATLAB simulations. The
classical fourth order R-K method with time-step 8
10h −
= has been used to solve the hyperchaotic
systems (40) and (41) with the adaptive controller defined by (44) and parameter update law
defined by (51).
The feedback gains in the adaptive controller (44) are taken as 5, ( 1, ,13).ik i= = 
The parameters of the hyperchaotic Wang and Li systems are taken as in the hyperchaotic case,
i.e.
40, 1.7, 88, 3, 35, 3, 12, 7, 0. 58a b c d r   = = = = = = = = =
For simulations, the initial conditions of the drive system (40) are taken as
1 2 3 4(0) 12, (0) 34, (0) 31, (0) 14x x x x= = − = =
Also, the initial conditions of the response system (41) are taken as
1 2 3 4(0) 25, (0) 18, (0) 12, (0) 29y y y y= − = = − =
Also, the initial conditions of the parameter estimates are taken as
ˆ ˆˆ ˆ(0) 21, (0) 14, (0) 26, (0) 16
ˆ ˆˆ ˆ ˆ(0) 17, (0) 22 (0) 15, (0) 11, (0) 7
a b c d
r   
= = = − =
= = − = = = −
Figure 9 depicts the anti-synchronization of the hyperchaotic Wang and hyperchaotic Li systems.
Figure 10 depicts the time-history of the anti-synchronization errors 1 2 3 4, , , .e e e e
Figure 11 depicts the time-history of the parameter estimation errors , , , .a b c de e e e
Figure 12 depicts the time-history of the parameter estimation errors , , , , .re e e e e   

17. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013
47
Figure 9. Anti-Synchronization of Hyperchaotic Wang and Hyperchaotic Li Systems
Figure 10. Time-History of the Anti-Synchronization Errors 1 2 3 4, , ,e e e e

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Figure 11. Time-History of the Parameter Estimation Errors , , ,a b c de e e e
Figure 12. Time-History of the Parameter Estimation Errors , , , , re e e e e   
7. CONCLUSIONS
This paper has used adaptive control theory and Lyapunov stability theory so as to solve the anti-
synchronization problem for the anti-synchronization of hyperchaotic Wang system (2008) and
hyperchaotic Li system (2005) with unknown parameters. Hyperchaotic systems are chaotic
systems with two or more positive Lyapunov exponents and they have viable applications like
chemical reactions, neural networks, secure communication, data encryption, neural networks,
etc. MATLAB simulations were depicted to illustrate the various adaptive anti-synchronization
results derived in this paper for the hyperchaotic Wang and Li systems.

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Author
Dr. V. Sundarapandian earned his D.Sc. in Electrical and Systems Engineering from
Washington University, St. Louis, USA in May 1996. He is Professor and Dean of the
R & D Centre at Vel Tech Dr. RR & Dr. SR Technical University, Chennai, Tamil
Nadu, India. So far, he has published over 300 research works in refereed
international journals. He has also published over 200 research papers in National and
International Conferences. He has delivered Key Note Addresses at many
International Conferences with IEEE and Springer Proceedings. He is an India Chair
of AIRCC. He is the Editor-in-Chief of the AIRCC Control Journals – International
Journal of Instrumentation and Control Systems, International Journal of Control
Theory and Computer Modeling, International Journal of Information Technology, Control and
Automation, International Journal of Chaos, Control, Modelling and Simulation, and International Journal
of Information Technology, Modeling and Computing. His research interests are Control Systems, Chaos
Theory, Soft Computing, Operations Research, Mathematical Modelling and Scientific Computing. He
has published four text-books and conducted many workshops on Scientific Computing, MATLAB and
SCILAB.